Image Processing Reference
In-Depth Information
This should be contrasted to progressing arithmetically, with the increment of the cell
size
Ω
, see Fig. 9.13. The explanation lies in the coordinate transformation, which
does not introduce new filter values (heights), but rather stretches the
ω
-axis. The t-
domain versions of these filters are found by inverse Fourier transforming
G
n
(
ξ
(
ω
))
w.r.t.
ω
. It is possible to show that functions sampled on nonregular lattices can be
reconstructed from their samples [5]. In consequence, the discretization discussed
here is a representation of the local spectrum.
In 2D, we wish to have a geometric progression of the tune-on frequencies in
the radial direction, whereas we wish to have an arithmetic progression of them in
the angular direction. The latter is easily justifiable because, a priori, all directions
appear to have equal importance in human vision as well as in numerous applications
in machine vision. Assuming that
is the 2D Cartesian coordinate vector in the
Fourier domain, the transformation that has the desired properties is given by the
log-polar coordinates:
ω
ξ
=
log
ω
max
ω
min
−
1
log
ω
ω
min
·
(9.51)
1
π
tan
−
1
(
η
=
ω
)
(9.52)
where
ξ
varies between [0
,
1] to the effect that
ω
varies between [
ω
min
,ω
max
], and
tan
−
1
(
1
,
1] for
η
.
We divide
ξ
,
η
uniformly as in Fig. 10.28 and place Gaussians in the marked cell
centers (
ξ
n
,η
n
):
G
n
(
ξ, η
)=exp
ω
) varies between [
−
π, π
], yielding a variation between [
−
exp
−
|
ξ
−
ξ
n
|
2
2
σ
ξ
−
|
η
−
η
n
|
2
2
σ
η
·
(9.53)
The radial part of this filter function was previously suggested in [137] when design-
ing quadrature mirror filters. Note that only one half of the plane needs to be covered
because we assume that the filter bank will be used to analyze real-valued signals
having
F
(
)=
F
∗
(
ω
−
ω
) only. The filters,
G
n
(
ξ
(
ω
)
,η
(
ω
)) can be readily sam-
) are available via Eqs.
(9.51)-(9.52). We show the isocurves of the resulting filters at the cell-boundaries for
seven frequencies and seven directions in Fig. 9.15 and mark the
tune-on frequen-
cies
by dots. The radial cross-section of the filters through the tune-on frequencies is
shown in Fig. 9.14.
The extension of the approach to
N
D Gabor filter banks is possible, provided
that the type of resource allocation scheme is at least roughly known. The latter is
crucial because it determines the coordinate transformations, which are not always
intuitively derivable. For an example, we refer to [21], where Gabor filters on an
irregular grid in 3D are suggested. The purpose is to mimic the speed sensitivity
of humans to apparent motion in image sequences. For example, using coordinate
transformations one can obtain a decomposition that is specially sensitive to low
spatial frequencies moving quickly and high frequencies moving slowly as compared
to other speed, and spatial frequency combinations.
pled on the original
Cartesian grid
ω
l
because
ξ
(
ω
) and
η
(
ω
